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General


This course is part of the MAGIC core.

Description

This couse provides an introduction to analysis in infinite dimensions with a minimum of prerequisites. The core of the course concerns operators on a Hilbert space including the continuous functional calculus for bounded selfadjoint operators and the spectral theorm for compact normal operators. There will be an emphasis on positivity and on matrices of operators. The course includes some basic introductory material on Banach spaces and Banach algebras. It also includes some elementary (infinite dimensional) linear algebra that is usually excluded from undergraduate curricula. Here is a very brief list of the many further topics that this course anticipates. C*-algebras, von Neumann algebras and operator spaces (which may be viewed respectively as noncommutative topology, noncommutative measure theory and `quantised' functional analysis); Hilbert C*-modules; noncommutative probability (e.g. free probability), the theory of quantum computing, dilation theory; unbounded Hilbert space operators, one-parameter semigroups and Schrodinger operators. And that is without starting to mention Applied Maths, Engineering and Statistics applications ...
Some relevant books. (See the Bibliography page for more details of these books.)
G. K. Pederson, Analysis Now (Springer, 1988)
[This course may be viewed as a preparation for studying this text (which is already a classic).]
Simmonds, Introduction to Topology and Modern Analysis (McGraw-Hill, 1963)
[Covers far more than the course, but is still distinguished by its great accessibility.]
P.R. Halmos, Hilbert Space Problem Book (Springer, 1982)
[Collected and developed by a master expositor.]
There are many many other books which cover the core part of this course.

Semester

Autumn 2018 (Monday, October 8 to Friday, December 14)

Hours

  • Live lecture hours: 20
  • Recorded lecture hours: 0
  • Total advised study hours: 80

Timetable

  • Mon 11:05 - 11:55
  • Tue 10:05 - 10:55

Prerequisites

Standard undergraduate linear algebra and real and complex analysis, and basic metric space/norm topology.

Syllabus

I PRELIMINARIES
Linear Algebra.
Including quotient space and free vector space constructions, diagonalisation of hermitian matrices, algebras, homomorphisms and ideals, group of units and spectrum.
Metric Space.
Review of basic properties, including completeness and extension of uniformly continuous functions.
General Topology.
Including compactness and Polish spaces.
Banach Space.
Including dual spaces, bounded operators, bidual [and weak*-topology], completion and continuous (linear) extension.
Banach Algebra.
Including Neumann series, continuity of inversion, spectrum, C*-algebra definition.
Hilbert Space Geometry.
Including Bessel's inequality, dimension, orthogonal complementation, nearest point projection for nonempty closed convex sets.
Stone-Weierstrass Approximation Theorem.
II HILBERT SPACE AND ITS OPERATORS
Sesquilinearity, orthogonal projection; Riesz-Frechet Theorem, adjoint operators, C*-property; Kernel-adjoint-range relation.
Finite rank operators; Operator types: normal, unitary, selfadjoint, isometric, compact, invertible, nonnegative, uniformly positive and partially isometric; Fourier transform as unitary operator; Invertibility criteria; Key examples of operators, finding their spectra (shifts and multiplication operators), norm and spectrum for a selfadjoint operator; Polar decomposition.
Continuous functional calculus for selfadjoint operators, with key examples: square-root and positive/negative parts; Matrices of operators, positivity in B(h+k);
Operator space - definition and simple examples;
Nonnegative-definite kernels, Kolmogorov decomposition; Hilbert space tensor products; Hilbert-Schmidt operators. Topologies on spaces of operators (WOT, SOT, uw). Compact and trace class operators, duality; Spectral Theorem for compact normal operators.
APPENDICES
Nets and generalised sums.
Topological vector spaces.

Lecturer


Martin Lindsay
Email j.m.lindsay@lancaster.ac.uk
Phone (01524) 594532
Photo of Martin Lindsay


Bibliography


Analysis nowG. K. Pedersen
Introduction to topology and modern analysisG.F. Simmons
A Hilbert space problem bookHalmos


Note:

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Assessment



No assessment information is available yet.

Functional Analysis

Files:
Released: Sunday 6 January 2019 (22.7 days to go)
Deadline: Sunday 20 January 2019 (37.7 days to go)
Instructions:

Assessment for FUN by MAGIC (MAGIC061).

Assessment for this course will be in the standard format for MAGIC, namely a take-home exam in January, with 2 weeks to complete, and submission via the MAGIC website. (Dates below.)
The Exam will consist of a mixture of routine-type questions and more challenging ones. You are encouraged to attempt the hardest ones that you can, but it will be possible to pass the exam with clear and correct solutions to straightforward questions. Some of the harder ones might need some mulling over ...
The best preparation for the Exam is to regularly do the Exercises given. It would be good to do some every week, attempting the hardest that you can.
Feedback, along with model solutions, will be distributed individually.
Good luck, in advance. Martin Lindsay
MAGIC061 Functional Analysis Exam, January 2019
To be released: Sunday 6 January 2019 Deadline: Sunday 20 January 2019



Recorded Lectures


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